Dear Geng Ouyang,

one possible answer is the following:

What is meant by 1 + 1/2 + 1/3 + 1/4 + ...

It is the limes of the sequence

1

1+ 1/2

1 + 1/2 + 1/3

...

Each of the elements of this sequence is a finite sum. And therefore in every pair of brackets you can place there can only be finitely many summands. Perhaps this is not an answer to the question you intended, but I was not yet able to read the papers I found on your profile.

Best regards

Johann Hartl

 It was Aristotle who first introduced a clear distinction to help make sense of it. He distinguished between two varieties of infinity. One of them he called a potential infinity. The potential infinite is a group of numbers or group of “things” that continues without terminating, going on or repeating itself over and over again with no recognizable ending point. What distinguishes the potential infinite and gives it the characteristic of being “potential” is the idea of grouping of natural numbers. No matter where you are while listing or counting out natural numbers, there always exists another number to precede the one before.

The actual infinite involves never-ending sets or “things” within a space that has a beginning and end; it is a series that is technically “completed” but consists of an infinite number of members. According to Aristotle, actual infinities cannot exist because they are paradoxical. It is impossible to say that you can always “take another step” or “add another member” in a completed set with a beginning and end, unlike a potential infinite. It is ultimately Aristotle’s rejection of the actual infinite that allowed him to refute Zeno’s paradox.

As we have discussed in an earlier thread of you, that actual infinity does not exist, of course, for calculations we can reach a working solution of a problem involving infinity by solving a few more steps so that obtained value is usable in our scientific or engineering endeavours, but never an absolutely precise value. But, such values may be potential not actual.

Yes, we do. when we write SUM 1/k! = e, we mean an actual infinity. When we write SUM 1/k = \infty, we mean a potential infinity. But when we write lim (1 + 1/n)^n = e we mean a potential infinity as well. So as you see, actual vs potential does not make any difference between infinities. It is only a qualitative difference when associating some semantic and meaning to those limits, which are similarly defined. 

Dear Dr. Arno Gorgels, Dr. Mohammad Firoz Khan and Dr. Mihai Prunescu, thank you for your view points.

We can see so many different ideas on actual infinite vs potential infinite, because they are unavoidably right there in our science theory system and have different definitions with different natures.

1, some people say mathematical analysis is potential infinite while some say no, it is actual infinite.

2, some people say theoretically and practically both actual infinite and potential infinite are really right there in our science while some say no-----they only acknowledge either theoretically or practically for either one.

3, some people say mathematical analysis is both actual infinite and potential infinite while some say no. Can we divide mathematical analysis into two parts: actual infinite mathematical analysis, potential infinite mathematical analysis?

4, people have been talking about actual infinite and potential infinite at least since Aristotle’s time; but not matter what and how much we talk, no one now in the world can answer following question: how many items each bigger than 1/2, or 100, or 1000000, or 1000000000000000,…can really be produced by the ‘brackets-placing rule’ from Un--->0 Harmonic Series in present classical “actual infinite” and “potential infinite” related mathematics? And, no one now in the world can tell what this case belongs to: actual infinite mathematical analysis or potential infinite mathematical analysis?

Best regards, 

Geng

Dear Dr. Arno Gorgels, thank you.

The problem is: “actual infinite and potential infinite” has nothing to do with countable or not (people never had countable or uncountable before set theory).

1, take the natural number set and real number set as examples, which is actual infinite or potential infinite in present classical “actual infinite and potential infinite” related mathematics?

2, does the above Harmonic Series case belong to actual infinite or potential infinite in present classical “actual infinite and potential infinite” related mathematics?

Best regards, 

Geng

Dear Dr. Arno Gorgels, thank you.

If natural number set and real number set are not either actual infinite or potential infinite, what kind of infinite do they belong to?

I have read many books about infinite; I know many people talk and talk about infinite, but few deal with practical infinite troubles in present classical “actual infinite and potential infinite” related mathematics.

So, talking about infinite is one thing and touching and challenging practical infinite troubles (suspended “actual infinite and potential infinite” related paradoxes) are another thing.

Best regards, 

Geng

 Dear Dr. Arno Gorgels, thank you.

Why the natural and real number sets are not potential infinite------natural number set is countable while real number set is uncountable?

Best regards, 

Geng

Dear Geng,

Natural numbers are not countable, there is always scope of adding 1 to previous number. As for as real numbers are concerned, take example of a single number like pi. As you calculate you will calculate it forever, numbers and groups of numbers after decimal will repeat again and again. There is no ending in the offing. However, to use it in our scientific calculations we will use a rounded figure of pi that gives us sufficient accurate results in the end. But, it is not actual infinite value. It will be a potentially infinite value. As, I have pointed out that infinity is a metaphysical (mother of science or at best physical science) or religious concept as evident from Pascal's wager:

Pascal's wager is an argument that asserts that one should believe in God, even if God's existence cannot be proved or disproved through reason.

Pascal's original wager is rooted in the belief the best course of action is to believe in God regardless of any lack of evidence, because that option gives the biggest potential gains. Pascal's original argument can be broken in four short steps:

1. If you believe in God and God does exist, you will be rewarded with eternal life in heaven: thus an infinite gain.

2. If you do not believe in God and God does exist, you will be condemned to remain in hell forever: thus an infinite loss.

3. If you believe in God and God does not exist, you will not be rewarded: thus a finite loss.

4. If you do not believe in God and God does not exist, you will not be rewarded, but you have lived your own life: thus a finite gain.

Dear Dr. Mohammad Firoz Khan, thank you.

But I have 2 questions:

1, in present classical “infinite” related mathematics, if “natural numbers are not countable”, what kind of number set is countable?

2,  if not all, what kind of concepts in human science are “metaphysical (mother of science or at best physical science) or religious concepts” and how can people distinguish?

Thank you again for your view points.

Best regards, 

Geng

Dear Geng,

Apart from religious discourse whether God exists or not, human curiosity about self (who am I? From where have I come? etc.) and about phenomena, things and objects that fill space similar questions and about their relations with being resulted in speculation about them.  They also experienced some beings and things are bad and harmful some are good and beneficial. From where we have come led people think about God and good and bad led to benefactor gods and those gods who bring harm etc. This concept is clear in Zoroastrianism in the form of God of virtue (Yazdan) and God of evil (Ahraman). This symmetry is represented in Greek, Roman and Ancient Iranian architecture and philosophy and found way in present day physics. In the same way metaphysical thinking about God, time and space gave birth to infinity.

Dear Dr. Mohammad Firoz Khan, thank you.

Dear Geng Ouyang,

the word infinity has different meanings. If in set theory you look at the set of natural numbers or the set of real numbers, these sets are infinite in the sense of actual infinity. The definition of infinity in this connection: A set M is infinite, if there is a mapping of M into M which is injective but not surjective.

If you have an infinite sum of for example real numbers, you have potential infinity. The infinite sum is not really a sum of infinitely many real numbers. It is defined as a limes of a series of finite sums. If it were otherwise, you could sum up the series 1 - 1/2 + 1/3 - 1/4 + ... as (1 + 1/3 + 1/5 + ... ) - (1/2 + 1/4 + ...), and that is not possible.

As I am not able to type the well-known symbol for infinity, I will write u instead. The formulas u + x = x + u = u for all real numbers x have to be interpreted using potential infinity. You cannot say what is u - u.

Best regards,

Johann Hartl

Dear Dr. Johann Hartl , thank you.

But I have 2 questions:

1, do you mean there are many different definitions with different natures  for “infinite” (at least two-------“actual infinite” and “potential infinite”) in mathematics?

2, Let’s see following divergent proof of Harmonic Series which can still be found in many current higher mathematical books written in all kinds of languages, which is a typical example of confusing “potential infinite” and “actual infinite”:

1＋1/2 ＋1/3＋1/4＋．．．＋1/n ＋．．．                                     （1）

=１＋1/2 ＋（1/3＋1/4 ）＋（1/5＋1/6＋1/7＋1/8）＋．．．        （２）

>1+ 1/2 ＋( 1/4＋1/4 )+（1/8＋1/8＋1/8＋1/8）＋．．．                  （3）

=1+ 1/2 + 1/2 + 1/2 + 1/2 + ．．．------>infinity                               （4）

Each operation in this proof is really unassailable within present classical “actual infinite--potential infinite” related science theory system. But, the unavoidable practical trouble is how many items in （1） can be added up by the “brackets-placing rule" to produce the infinite items in （3） each bigger than 1/2 or 1 or 100 or 100000 or 1000000000000000 or…?

 “Potential infinite” or “actual infinite” decides whether or not we can produce infinite items each bigger than 1/2 or 1 or 100 or 100000 or 1000000000000000 or… from Un--->0 infinite Harmonic Series by “brackets-placing rule".

This is the newly discovered Harmonic Series Paradox-------a modern version of 2500-year old suspended Zeno's Achilles--Turtle Race Paradox.

Thank you again for your view points.

Best regards, 

Geng

Dear Geng Ouyang,

one possible answer is the following:

What is meant by 1 + 1/2 + 1/3 + 1/4 + ...

It is the limes of the sequence

1

1+ 1/2

1 + 1/2 + 1/3

...

Each of the elements of this sequence is a finite sum. And therefore in every pair of brackets you can place there can only be finitely many summands. Perhaps this is not an answer to the question you intended, but I was not yet able to read the papers I found on your profile.

Best regards

Johann Hartl

Dear Dr. Johann Hartl , thank you.

What I talk about in above post is the divergent proof of Harmonic Series which can still be found in many current higher mathematical books written in all kinds of languages;  this newly discovered Harmonic Series Paradox has been cited in five of my uploaded 10 papers as a typical example of the fatal confusing and mixing up errors of “potential infinite” and “actual infinite” in the foundation of present classical infinite related philosophy and mathematics. Do you mean in your above post that it is impossible to produce infinite items in （3） each bigger than 1/2 or 1 or 100 or 100000 or 1000000000000000 or…by the “brackets-placing rule"? This is the very paradoxical point: some prove yes (most people prove yes) and some prove no (such as your proof on your above post).

And, one question has been troubling us human for at least 2500 years since Zeno's time: can we have only one definition for “infinite” or many different definitions with different natures for “infinite” in present classical infinite related philosophy and mathematics? What is your idea dear Dr. Johann Hartl?

Best regards, 

Geng

I intend to make following addition to you discussion.

1. The set of Natural Numbers according to Cantor is countable!

2. It's right that the notions of Actual and Potential infinities were introduced by Aristotle but in his opinion Actual infinity could not exist. In my opinion Cantor disregarded Aristotle's point of view. Let's take the Set of Natural numbers as an example.

When we write Natural Numbers as 1,2,3... we are treating it as a Potential Infinity as this process in unending. But the moment we write it as N={1,2,3,...} which means that we are looking at "ALL" natural numbers at once and hence it's an Actual Infinity. 

Dear Dr. Z. Shareef , thank you.

We can really see from our infinite related science history that many people understand “Actual and Potential infinities” in all kinds of good ways. That is ok.

But, how to deal with those items of Un--->0 in Harmonic Series (actual or potential infinite: with or without quantitative-value 0f Un--->0) decides whether or not we can produce infinite items each bigger than 1/2 or 1 or 100 or 100000 or 1000000000000000 or… from Un--->0 infinite Harmonic Series by “brackets-placing rule".

If those items of Un--->0 are treated as “with- quantitative-value Un--->0”, we can produce infinite items each bigger than 1/2 or 1 or 100 or 100000 or 1000000000000000 or… from Un--->0 infinite Harmonic Series by “brackets-placing rule"; if those items of Un--->0 are treated as “without-quantitative-value Un--->0”, we can not produce infinite items each bigger than 1/2 or 1 or 100 or 100000 or 1000000000000000 or… from Un--->0 infinite Harmonic Series by “brackets-placing rule".

Your idea on the above “actual or potential infinite: with or without quantitative-value  0f Un--->0” would be very much appreciated. Thank you again.

Best regards, 

Geng

Dear Shareef

Of course, Cantor disregarded Aristotle's point of view. But, what happen to him in search of infinite sets of real numbers. Please go through last paragraph of the link below:

http://www.storyofmathematics.com/19th_cantor.html

For a detail discussion on infinity please see post:

https://www.researchgate.net/post/Does_infinity_exist_or_it_is_a_product_of_human_mind

But it is very pity that working in present classical philosophy and mathematics basing on “potential infinite” and “actual infinite”, most people can not run away from being dominated by the mistakenly confusing of “potential infinite” and “actual infinite”.

The infinite small (few) related errors and paradox families (such as the newly discovered Harmonic Series Paradox) and the infinite big (many) related errors and paradox families (such as the newly discovered Cantor’s ideas and operating process of mistaken diagonal proof of “the elements in real number set are more infinite than that in natural number set”) are typical examples of “master pieces of confusing potential infinite and actual infinite”.